The Earth's hydrogen exosphere Lyman‐α radiation was mapped with the Solar Wind Anisotropies/Solar and Heliospheric Observatory (SWAN/SOHO) instrument in January 1996, 1997, and 1998 (low solar activity). The use of a hydrogen absorption cell allowed to disentangle the interplanetary emission from the geocoronal one and to assign the absorbed signal almost entirely to the geocorona. The geocorona was found to extend at least up to 100 Earth radii (RE) with an intensity of 5 Rayleigh, an unprecedented distance well exceeding the recent results of Lyman Alpha Imaging Camera (LAICA) imager (∼50 RE), and encompassing the orbit of the Moon (∼60 RE). We developed a numerical kinetic model of the hydrogen atoms distribution in the exosphere, which includes the solar Lyman‐α radiation pressure and the ionization. The radiation pressure compresses the H exosphere on the dayside, producing a bulge of H density between 3 and 20 RE, which fits observed intensities very well. The SWAN Lyman‐α distribution of intensity was compared both to LAICA (2015) and to Orbiting Geophysical Observatory number 5 (1968) measurements. Integrated H densities of SWAN at a tangent distance of 7 RE are larger than LAICA/Orbiting Geophysical Observatory number 5 by factors 1.1–2.5, while we should expect a stronger effect of the radiation pressure at solar max. We discuss the possible role of H atoms in satellite orbits to explain this apparent contradiction. An onion‐peeling technique is used to retrieve hydrogen number density in the exosphere for the three SWAN observations. They show an excess of density versus models at large distances, which is likely due to nonthermal atoms (not in the model). Key Points We find that the geocorona extends to almost twice the distance of the Moon the H exosphere is compressed by solar radiation pressure, forming a bulge on the dayside this bulge is enhanced at low solar activity, possibly in relation with a population of Hatoms in satellite orbits